Georgia Institute of Technology College of Sciences

This thesis introduces a modular framework written in Macaulay2 designed to solve nonlinear algebra problems.  First, we will introduce the background for the framework, covering gates, circuits, and straight-line programs, and then we will define the gates used in the framework.  The remainder of the talk will include well-known algorithms such as Newton's method and Runge-Kutta for solving nonlinear algebra problems, their implementation in the framework, and explicit conic problems with a comparison between different methods.

We develop a unified framework for improving numerical solvers with Neural Networks with Locally Converging Inputs (NNLCI). First, we applied NNLCI to 2D Maxwell’s equations with perfectly matched‐layer boundary conditions for light–PEC (perfect electric conductor) interactions. A network trained on local patches around specific PEC shapes successfully predicted solutions on globally different geometries. Next, we tested NNLCI on various ODEs: it failed for chaotic systems (e.g., double pendulum) but was effective for nonchaotic dynamics, and in simple cases can be interpreted as a well‐defined function of its inputs. Although originally formulated for hyperbolic conservation laws, NNLCI also performed well on parabolic and elliptic problems, as demonstrated in a 1D Poisson–Nernst–Planck ion‐channel model. Building on these results, we applied NNLCI to multi‐asset cash‐or‐nothing options under Black–Scholes. By correcting coarse‐ and fine‐mesh ADI solutions, NNLCI reduced RMSE by factors of 4–12 on test parameters, even when trained on a small fraction of the parameter grid. Careful treatment of far‐field boundary truncation was critical to maintain convergence far from the strike price. Finally, we demonstrate NNLCI’s first application to quantum algorithms by improving variational quantum‐algorithm (VQA) outputs for the 1D Poisson equation under realistic NISQ‐device noise. Although noisy VQA solutions deviate from classical finite‐difference references and do not converge to true solutions, NNLCI effectively maps these noisy outputs toward high‐accuracy references. We hypothesize that NNLCI implicitly composes the map from coarse quantum outputs to a noisy convergence space, then to the true solution. We discuss conditions for NNLCI to approximate a well‐defined inverse of the numerical scheme and contrast this with Monte Carlo methods, which lack deterministic intermediate states. These results establish NNLCI as a versatile, data‐efficient tool for accelerating solvers in classical and quantum settings.

After quickly recalling the established theory on the combinatorics of orthogonal matroids, we define and study basic properties of the extended rank function and the modular tuples in orthogonal matroids. We then prove a weak version of the path theorem concerning the connectivity of circuits. 

Next, we consider representations of orthogonal matroids over fields (and more generally, over tracts) by bases. We then give a few applications, purely using this basis approach, to the representation theory of orthogonal matroids. We also give a different way of representing orthogonal matroids by circuit functions, which is proved to be equivalent to the basis approach. This is based on joint work with Matthew Baker and joint work with Donggyu Kim. 

The final part of the thesis focuses on the rescaling classes of representations. We construct the foundation of an orthogonal matroid, which possesses the universal property that the set of rescaling classes of representations is in one-to-one correspondence with the set of morphisms from the foundation to the target field. We also give explicit generators and relations of the foundation and an algorithm for computations. 

Zoom link: https://gatech.zoom.us/my/tongjinmath?pwd=QzRDalp2ditGL2tVNUozWm1RK1UwUT09

This thesis adopts the immersed-curve perspective to analyze the knot Floer complexes of (1,1)-satellite knots. The main idea is to encode the chain model construction through what we call a planar (1,1)-pairing. This combinatorial and geometric object captures the interaction between the companion and the pattern via the geometry of immersed and embedded curves on a torus (or its planar lift). By working with explicitly constructed (1,1)-diagrams and their planar analogs, we derive rank inequalities for knot Floer homology and develop a geometric algorithm for computing torsion orders. The latter, based on a depth-search procedure, translates intricate algebraic operations into tangible geometric moves on planar (1,1)-pairings, further yielding results on unknotting numbers and fusion numbers.